$ D = \left[\begin{array}{rr}1 & 2 \\ -2 & 0\end{array}\right]$ $ E = \left[\begin{array}{rrr}0 & 1 & 1 \\ 2 & -1 & 4\end{array}\right]$ What is $ D E$ ?
Because $ D$ has dimensions $(2\times2)$ and $ E$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ D E = \left[\begin{array}{rr}{1} & {2} \\ {-2} & {0}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{1} & \color{#9D38BD}{1} \\ {2} & \color{#DF0030}{-1} & \color{#9D38BD}{4}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{0}+{2}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{0}+{2}\cdot{2} & ? & ? \\ {-2}\cdot{0}+{0}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{0}+{2}\cdot{2} & {1}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{-1} & ? \\ {-2}\cdot{0}+{0}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{0}+{2}\cdot{2} & {1}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{-1} & {1}\cdot\color{#9D38BD}{1}+{2}\cdot\color{#9D38BD}{4} \\ {-2}\cdot{0}+{0}\cdot{2} & {-2}\cdot\color{#DF0030}{1}+{0}\cdot\color{#DF0030}{-1} & {-2}\cdot\color{#9D38BD}{1}+{0}\cdot\color{#9D38BD}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}4 & -1 & 9 \\ 0 & -2 & -2\end{array}\right] $